################## The angle sum rule ################## The angle sum rule is: .. math:: \sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta ***** Proof ***** Let's say we have a vector $(x_1, y_1)$ resulting from the anticlockwise rotation of a length 1 vector $(1, 0)$ by $\alpha$ degrees around the origin. We rotate this vector another $\beta$ degrees anticlockwise around the origin to give length 1 vector $(x_2, y_2)$. .. image:: images/angle_sum.png We can see from the picture that: .. math:: \cos(\alpha + \beta) = x_2 = r - u \sin(\alpha + \beta) = y_2 = t + s We are going to use some basic trigonometry to get the lengths of $r, u, t, s$. Because the angles in a triangle sum to 180 degrees, $\phi$ on the picture is $90 - \alpha$ and therefore the angle between lines $q, t$ is also $\alpha$. Remembering the definitions of $\cos$ and $\sin$: .. math:: \cos\theta = \frac{A}{H} \implies A = (\cos \theta) H \sin\theta = \frac{O}{H} \implies O = (\sin \theta) H Thus: .. math:: p = \cos \beta q = \sin \beta r = (\cos \alpha) p = \cos \alpha \cos \beta s = (\sin \alpha) p = \sin \alpha \cos \beta t = (\cos \alpha) q = \cos \alpha \sin \beta u = (\sin \alpha) q = \sin \alpha \sin \beta So: .. math:: \cos(\alpha + \beta) = x_2 = r - u = \cos \alpha \cos \beta - \sin \alpha \sin \beta \sin(\alpha + \beta) = y_2 = t + s = \sin \alpha \cos \beta + \cos \alpha \sin \beta .. include:: links_names.inc